Direct detection of photoinduced magnetic force at the nanoscale reveals magnetic nearfield of structured light

We demonstrate experimentally the detection of magnetic force at optical frequencies, defined as the dipolar Lorentz force exerted on a photoinduced magnetic dipole excited by the magnetic component of light. Historically, this magnetic force has been considered elusive since, at optical frequencies, magnetic effects are usually overshadowed by the interaction of the electric component of light, making it difficult to recognize the direct magnetic force from the dominant electric forces. To overcome this challenge, we develop a photoinduced magnetic force characterization method that exploits a magnetic nanoprobe under structured light illumination. This approach enables the direct detection of the magnetic force, revealing the magnetic nearfield distribution at the nanoscale, while maximally suppressing its electric counterpart. The proposed method opens up new avenues for nanoscopy based on optical magnetic contrast, offering a research tool for all-optical spin control and optomagnetic manipulation of matter at the nanoscale.


Si probe on top of glass substrate illuminated by an APB
As the essential part of the proposed photoinduced magnetic force detection method, in this subsection we discuss a physical modeling for the Si probe on top of the glass substrate illuminated by an incident APB.
As has already been discussed in a previous work (24), a subwavelength Si nanoparticle can be considered as a pure magnetic dipole scatterer provided that a proper symmetric excitation is applied to exclusively excite its magnetic dipole moment. In that work, we have experimentally obtained the force map, based on the electric field component of light, exerted on a gold tip on top of a Si truncated cone located on a glass substrate and illuminated by an APB. Therefore, in that work we measured the electric dipolar force exerted on the scanning nanotip (that is an electric nanoprobe). We have shown that the measured force is proportional to the incident electric field intensity, and that is the reason we call it the electric force of light (23). Based on experimental results and on rigorous theoretical analysis based on the calculation of multipole moments, in (24) we have indirectly demonstrated (from the measurement of the electric force) that a Si disk is an effective magnetic scatterer at a specific wavelength range and under APB excitation.
Here we use the magnetic scatterer investigated in (24) as a probe of the magnetic force, and devise a new force microscopy technique which enables direct and exclusive acquisition of the optical force due to the magnetic field component of light rather than its electric counterpart. As noted in the main text of the manuscript, this method enables the direct measurement of magnetic field component of light, in contrast of the common PiFM system that only directly detects the electric field component of light.
Considering the dipole approximation of the Si magnetic nanoprobe of the proposed system shown in Fig. S1, the validity of which is provided in the following, the simplest scenario that mimics the interaction between the Si probe (as a dipole scatterer) and the glass substrate under APB illumination is shown in Fig. S1 (Right). First, we replace the Si magnetic nanoprobe with a Si sphere that carries a Mie "magnetic" resonance, then we replace the overall effect of the substrate with an image dipole. As a result, we consider two closely spaced Si sphere that mainly provide two coupled magnetic dipoles; one representing the probe whereas the other illustrating its image. In the following we also explore the role of the other contributions like the electric dipole and quadrupoles, demonstrating the dominance of the magnetic dipole in generating the total force on the nanoprobe. Fig. S1 illustrates these two scenarios, i.e., the real scenario of the Si probe on top of the glass substrate, and the equivalent scenario of the two identical spheres illuminated by an APB from the bottom (42).
The equivalent problem of two interacting magnetic dipolar dipoles is treated analytically and it provides a basic physical insight of the real scenario. During the whole analysis, we keep in mind that the main goal is to find the optical force exerted on the Si probe on top of the substrate when illuminated by an APB from the bottom (see Fig. S1).

Fig. S1. Real scenario Left) and its physical modeling Right).
The Si probe on the top of a glass substrate is illuminated with an APB incident from the bottom side. In the physical modeling, the truncated cone shape of the probe is modeled as a sphere shape for the convenience of analytical derivation. And the substrate is replaced by the image sphere to mimic the electromagnetic interaction between the probe and the substrate.

The Lorentz force density and the total optical force
The optical force on an object is derived from the Lorentz force density generated by a distribution of charge and current densities and , respectively. The Lorentz force density exerted at any position is given by where is the electric field vector, and is the magnetic induction vector. Here all italic symbols define position and time dependent quantities. Therefore, the total force on the proposed object with volume containing these charge and current distributions reads . Since we do not consider any "impressed" (or forced) source within the magnetic nanoprobe volume in our scenario, then, the force is calculated using the induced charge and current densities, and , respectively in the magnetic nanoprobe, leading to . The total time-average force exerted on the probe is provided by volume integration, leading to where non-italic fonts denote phasors, * denotes complex conjugation, and we have implicitly assumed a time dependence. The volumetric integral (S2) and has been numerically calculated directly using COMSOL evaluated fields, to produce the results of in Figs. 2, S3, and S4. The term in Fig. 3 has been calculated in the same way, but using only the term in (S2).

Optical force and the dipole approximation
By neglecting all multipoles except the dipoles, one can model the nanoprobe as a dipolar scatterer with induced electric and magnetic dipole moments and , respectively. The timeaveraged optical force exerted on the object for time harmonic fields with time dependence is represented in terms of phasors as follows (13): The i-th component of the time-averaged optical force in Eq. (S3) reads, for Cartesian coordinates. Note that one may consider the effect of higher order multipoles and calculate the optical force exerted on the object as in Ref. (43), however, within an acceptable approximation range it is enough to only include the effect of dipole moments in our analysis, as demonstrated in our results when comparing the dipolar force to the total Lorentz force.

The system of coupled Si probe-image under APB illumination
Let us first demonstrate that the probe in the system of a coupled sphere under APB illumination is a magnetic scatterer at a specific wavelength range and under certain alignment condition. To that end, we calculate the multipoles (up to magnetic quadrupoles) of the top sphere (as the model of the probe) when it is coupled with the image sphere (that mimics the effect of the substrate). As the representation of the realistic fabricated probe, we design the probe and image spheres to have the comparable size with 92nm radius, and their magnetic resonance wavelength is around 610 nm. We prove that under a specific alignment between the two spheres and the beam, the probe sphere is exclusively a magnetic dipole scatterer. Indeed, we calculate and compare the power scattered by each multipole and show that under such an alignment the scattered power due to the magnetic dipole of the probe sphere is dominant. Accordingly, we consider only the first four multipoles, i.e., electric and magnetic dipole as well as quadrupole moments , , , and , respectively, which are enough for our analysis proof due to the small size of the scatterer is the diameter of the Si sphere and is the magnetic resonance wavelength for this radius). The electric and magnetic dipole moments and are two vectors whereas the electric and magnetic quadrupole moments and are two tensors of second rank. The components of each moment read (42,44,45): respectively. Here, in Cartesian coordinates, and are , and coordinates and such indices are used to represent Cartesian components of vectors and tensors, whereas is the position vector and . Moreover, is the Kronecker delta, and are, respectively, the charge and current density distributions over the scatterer volume V, and the integrations are taken over the entire volume of the scatterer. It can be shown that the scattered power due to these multipoles reads (42,46): Here and are angular frequency and wavenumber whereas and are the free space permittivity and permeability, respectively. Fig. S2 demonstrates the contribution to the total scattered power, Eq. (S6), generated by the Si probe sphere, provided by the electric and magnetic dipoles, and by the electric and magnetic quadrupoles defined in Eq. (S5). Results are obtained using full-wave numerical calculations carried out with COMSOL Multiphysics software which is based on the finite element method (33) (FEM).
In Fig. S2 we plot the multipole contributions to the total scattered power given by Eq. (S6) for the probe sphere by moving the system of two spheres from the center of the beam axis (at x = 0 nm) toward the edge of the beam waist (x = 450 nm). The two spheres have a radius of 92nm radius nm, and the gap between them is 5 nm. In these calculations, we have used an APB with beam waist parameter and carrying a power of 150 µW coming from the bottom, with its minimum waist at z = 0. As it is clear from this figure, the scattering power due to the magnetic dipole is dominant when the system of two spheres is fully aligned with the axis of the excitation beam (at x=0). In such a scenario, the total scattered power peaks at which corresponds to its magnetic resonance. Interestingly, the footprint of the magnetic dipole always presents even when the nanoprobe is laterally displaced with respect to the beam axis when the scattering of the electric dipole is dominant.

The exerted force on the Si probe sphere in the presence of its image sphere under the APB illumination
We have already proven in the previous section that scattering by the probe Si sphere at the presence of the image is dominated by the scattering of the magnetic dipole moment at the wavelength of interest, when illuminated by an APB. Here we calculate the exerted force on this magnetic Mie resonator integrating the Lorentz force density formulation of Eq. (S1), and demonstrate that the total force calculated using this force formulation on this magnetic resonator with a good approximation is equivalent to where, is partial derivative with respect to the i-th spatial coordinate, and i,j=x,y,z are the Cartesian coordinates. This corresponds to the components of the magnetic dipole force expression . Indeed, in the main body of the paper, this force was called as the magnetic dipolar force since it only includes the interaction between the magnetic dipole moment and the magnetic field gradient. In this paper, the magnetic dipole force in Eqs. (S7) and (3) is evaluated by the product of the numerically calculated magnetic dipole as in Eq. (S5) and the numerically calculated magnetic field gradient at the center of the tip sphere, by using the finite element method simulations implemented in COMSOL Multiphysics. When this formula is applied to the Si truncated cone tip, the gradient is evaluated at the center of the cone.
Fig . S3 shows the total optical force map, calculated using (S2), exerted on the top Si sphere (representing the tip of the probe) and its image sphere illuminated by an APB from the bottom. The two spheres system is the same as the one considered in the previous section, i.e., the have a radius of 92nm radius nm, and the gap between them is 5 nm. The force is calculated at each position when we laterally move the Si tip and image spheres, relative to the APB illumination axis for two wavelengths: 610 nm (magnetic resonance) and 550 nm (electric resonance). As shown in this figure, the force map shows a bright center circular shape with a hot spot at the magnetic resonance of the scatterer at 610 nm, and a doughnut shape at the electric resonance (550 nm) as it is also the case in the experimental measurements discussed in the main body of the manuscript. Importantly, the force map at the magnetic resonance resembles the magnetic field distribution as discussed in the main manuscript and will be shown in a next section here. Indeed, this figure proves that the magnetic force and the magnetic field are proportional as shown in Eq. (4) in the manuscript and is proven below. In Fig. S4 we compare the exerted force calculated from the Lorentz total force formulation in Eq. (S1) integrated over the whole top sphere with the contributions due to the electric and magnetic dipolar forces calculated from the dipole approximation formulation in Eq. (S3). The comparison is done at two wavelengths: at 610 nm, as the on-state condition (on magnetic resonance), and at 550 nm, as the off-state condition (off magnetic resonance).
The different terms in Fig. S4 are all longitudinal force components defined as follows: the time-averaged total optical force exerted on the nanoprobe is defined in Eq. (S2); the timeaverage magnetic dipolar force is defined in Eqs. (3) and (S7); the time-average second term in the Lorentz formula due to the magnetic field B is ; and the time-average electric dipolar force is defined in Eqs. (2). As shown in Fig. S4(a), the total force exerted on the tip is in close agreement with the magnetic dipolar force in a wide range of relative positions between the beam axis and the two coupled scatterers at the magnetic resonance wavelength whereas the agreement does not hold for the off-magnetic resonant wavelength shown in Fig. S4(b). This proves that the total optical force is purely magnetic within the interested wavelength range and under this specific type of excitation scenario.

Dependence of the total dipolar force on the sharp/blunt probe
Given the tapered shaft of the sharp tip, we note that a stronger optical response from the tip can be expected at locations higher up the shaft, as circulating currents associated with the Mie resonance are more prominent when the shaft diameter gradually increases. For sharper tips at the chosen excitation wavelength F z these currents move from the apex towards the base of the conical tip and that is why the dipole-dipole distance increases for sharper tips. A similar phenomenon has been reported in the literature (47,48). Since the photo-induced force decreases as 1/d 4 , where d is the distance between the sample and the location in the tip material, a meaningful force contribution from the tip material higher up the shaft (larger d) is suppressed. Consequently, the overall response of the sharp Si tip is expected to be much weaker than the response of the blunt Si tip.

Derivation of Eq. (4) of the main manuscript
Referring to Eq. (S7), the local magnetic field, possesses one contribution from the incident APB beam and the other one from the scattered beam due the reflections from the substrate.
Here by using the image principle (42), we replace the substrate with an image dipole, and consider its interactive effect in the field contributions from the image dipole. Therefore, the local magnetic field at the tip position reads and is the magnetic dyadic Green's function in Cartesian coordinates with being the position vector at the tip/image location. Due to optically small distance between the tip and its image, the near field term in the Green's function is dominant and is approximated by (42): is the identity tensor of rank 2 and is the unit vector from the source to the observation point. The dipole moments of the image dipole and the tip dipole are (S11) (S12) Here and are the dipolar magnetic polarizability tensors of the tip and image, respectively. Similarly, the local magnetic field at the image position reads (S13) where, Inserting Eq. (S14) in Eq. (S13) and then by using the resulting equation in Eq. (S11), we derive a closed-form expression for which is eventually used to obtain from Eqs.  Finally, by using Eq. (S12) in the above equation, we obtain (S16) where, and are the magnetic polarizability components of the tip and the image dipole, respectively, in Cartesian coordinates and is the vertical distance between the tip and the image dipole. In obtaining Eq. (S16), we considered that the tip and its image are positioned along the z-axis. Moreover, in determining the local magnetic fields, we neglected terms containing incident fields compared to terms containing the gradient of scattered fields since their values are much smaller in the near zone of a scatterer. Besides, we have ignored all the terms containing polarizability power orders higher than second and since the dipole and its image are optically very close, we have neglected the terms containing compared to those containing .
Next, if the phase difference of the incident beam between the tip and its image is neglected due to their deep subwavelength distance, we can assume that and the force reduces to (S17) If one considers a reference system such that the polarizability tensors are diagonal, i.e., , and also assume an azimuthally symmetric scatterer, i.e.,  proportional to the tip dipole , from Eq. (S18) the time-averaged optical force on the tip is related to the incident magnetic field at the tip position as which is Eq. (4) of the manuscript. Note that we have suppressed superscript "dipole" and subscript "m" in the notation of this section for simplicity.

Approximation condition of Eq. (4) with axis displacement for the real scenario with a truncated Si cone
We analyze the approximation condition of Eq. (4) in the main paper, i.e., the proportionality between the magnetic force and the longitudinal magnetic field intensity, even when the probe is slightly misaligned with respect to the axis of the incident APB. For this purpose, we define the normalized longitudinal magnetic field intensity to the maximum magnetic force as .
We perform the simulation of the realistic scenario as indicated in the main paper: the onstate truncated cone probe above the glass substrate is scanning in the transverse direction (x) around the axis of the incident APB, at 670 nm wavelength. The APB has a beam waist parameter of with incident beam power of 150 μW. We compare the normalized longitudinal magnetic field intensity and the longitudinal magnetic force with respect to the displacement x between the axis of the incident APB and the on-state probe, shown in Fig. S6. The light blue curve is the force due to induced magnetic dipole in the on-state truncated Si disk calculated by Eq. (5) whereas the dark blue curve is the longitudinal incident magnetic field intensity normalized to maximum force due to induced magnetic dipole.
We highlight the yellow region where the normalized longitudinal magnetic field and the magnetic force have an overlap accuracy (ratio) higher than 90%: the spot has a remarkable diameter close to 400 nm. Considering the wavelength for on-state magnetic excitation as 670 nm, the highlighted region shows a rather large area near APB axis where the proportionality between the longitudinal magnetic force and the incident magnetic field intensity holds.
The overlap accuracy indicated by the yellow highlight is related to the accuracy of the dipolar approximation upon a slight displacement between the axes of the incident beam and the probe. With a broken azimuthal symmetry, other multipoles rather than the desired magnetic dipole (i.e. electric dipole, electric/magnetic quadrupole etc) will be excited as shown in Fig S2,   The force is exerted on the magnetic nanoprobe (on-state Si truncated cone) 5nm over the dielectric substrate versus displacement x with respect to the axis of the incident APB. The yellow highlight indicates the region where the ratio between the normalized longitudinal magnetic field intensity and the magnetic force is greater than 90%. The magnetic dipole force is calculated using Eq. (3).

Comparison between gold-coated probe and the Si probe
As described in the main manuscript, both the gold-coated sharp probe and blunt Si tip can serve as the electric probe to measure the electric force by the PiFM system. This is because the permittivity of both gold and Si allows electric dipole excitation by the light source in the probe, which gives rise to a nonzero dipole-mediated electric force.
However, there are important differences as well. Gold has a negative real permittivity and a small but non-negligible imaginary permittivity (loss) in the optical region. While Si also has low loss, relative to gold it has a smaller and positive permittivity. Consequently, the electric dipolar polarizability of gold tips is expected to be much stronger than for Si, producing stronger PiFM signals and thus superior SNR in the force detection measurement compared to a Si tip. The excellent polarizability of a gold probe can also enable the generation of detectable PiFM signals from small diameter tips (<10nm or even 1nm). On the other hand, a sharp Si probe may not have enough effective polarizability to induce a measurable electric force. Therefore, to enable efficient electric force detection, the blunt apex is necessary for the Si probe to provide a strong enough dipole m, electric dipole that is also close enough to the sample/substrate surface. This will naturally decrease the achievable resolution by the blunt Si probe. We provide a comparison between a PiFM measurement performed with a sharp gold tip (diameter<10nm) and a blunt Si probe (diameter~200nm) for the case of a tightly focused APB, which shows clear differences in signal strength and SNR in Fig. S7.   Fig. S6. The force maps of tightly focused APB. They are detected by the Left) sharp gold probe and Right) blunt Si probe, respectively. The force map detected by the sharp gold probe is from our previously published work, reprinted with permission from ref. (24). Copyright 2018 American Chemical Society.